I am taking a course in Algebraic Topology next semester, so I thought of starting to read about it on my own.
My concern: I have read Topology by Munkres, and read the first chapter on algebraic topology from that book, so I have an idea about the fundamental group and covering spaces. Later I read the book Topology by Klaus Janich till the chapter on CW complexes. I really liked the book: it was not rigorous (somewhat unclear at some places) but enough to draw my interest in the subject. I was planning to start reading algebraic topology more seriously, so a senior suggested me the book Algebraic Topology: A First Course by William Fulton. I have read the first chapter, but I am having some problems. I actually don't have a good background in multivariable calculus. The author gives an overview of calculus in the plane and he introduces some basic concepts about multivariable calculus. I really like algebra; I have a good background in fields and Galois theory, ring and group theory (don't know free abelian groups yet). I am reading commutative algebra side by side. So I would really like to learn algebraic topology in an 'algebraic way', and I am concerned that Fulton in this book does this by another way (a differential approach or something). But at the same time I want to learn algebraic topology in an intuitive way (with pictures) and I've been told that this is the most expository book available on algebraic topology. Am I right about this book, I mean does it approach algebraic topology using category theory and other algebraic tools or in some other way? If so please tell me is it more important for intuition, so that I can read more on multivariable calculus and then start again?
I know this kind of question is not supported in this community, but I do need help about this. My course will be starting from next month, and I want to get a good concept about this subject. And I want to take up a course about algebraic geometry too. That's why I don't want to spend time experimenting. Thanks in advance.
Best Answer
A cursory glance through the book suggests it is has a strong differential flavour. If you’re familiar with multivariable calculus especially as formulated through differential forms and exterior derivatives, this isn’t necessarily a bad thing, and does help in understand what it means for two cycles to be homologous. The book presents de Rham cohomology even before homology, and restricts itself to the 0th and 1st (co)homology groups. It also briefly discusses vector fields which are more likely to be covered in a differential topology course. It includes Čech cohomology which is not something you meet in a first algebraic topology course usually.
So, I wouldn’t think that this book matches your background or aims. I’d recommend Topology & Geometry by Bredon as one that I liked, but there are multiple books out there that would serve you better! Feel free to reach out to me for more suggestions or recommendations on books.